Cross-linked pair of polymer chains under strong tension

TL;DR

This work analyzes two minimal cross-linked polymer systems under strong tension: a pair of chains sharing an endpoint connected by a harmonic cross-link, and a long chain pair organized as a necklace of reversible cross-links. Using weakly bending theories for freely jointed and wormlike chains, it derives analytic force–extension relations and transverse/longitudinal fluctuations, showing that a single cross-link leaves tensile elasticity unchanged in the thermodynamic limit while substantially suppressing transverse fluctuations, effectively forming a loop. For the necklace, a Gaussian slinky mapping casts the transverse conformation into a two-dimensional loop ensemble, while a continuum/quantum analogy for shallow binding wells reveals a force-driven crossover between weakly and strongly bound regimes with no true phase transition. The results unify discrete and semiflexible models, provide explicit scaling forms for binding and elasticity under force, and offer insights relevant to bundled biopolymers and reversible cross-linked networks.

Abstract

We study two cross-linked polymer systems in the strong stretching regime. The first consists of two polymers sharing one endpoint, with the other two endpoints coupled by a harmonic potential. Within the weakly bending approximation, we analyze the tensile elastic response for freely jointed or wormlike chains; for the latter, the approximation applies either at large tension or at moderate tension with large persistence length (rodlike limit). We obtain analytic expressions for the force--extension relation and for the longitudinal and transverse mismatch of the cross-linked endpoints. In the thermodynamic limit, the cross-link does not affect the tensile elasticity, but it significantly suppresses transverse fluctuations, effectively forming a loop structure. The second system is a polymer necklace in the thermodynamic limit, composed of two strongly stretched polymers interconnected by a regular sequence of reversible cross-links. Using an analogy with a two-dimensional system of concatenated Gaussian loops ("Gaussian slinky"), we calculate the mean fraction of cross-linked sites as a function of the tensile force and find weak and strong binding regimes connected by a crossover. For shallow binding potential wells (compared with $k_{\rm{B}}T$), we employ a continuum description and exploit the mapping between directed polymers and a two-dimensional quantum particle to determine the crossover behavior and the mean transverse separation between the two polymer chains.

Figures (8)

• Figure 1: Schematic diagram of the cross-linked freely jointed loop under strong tension.
• Figure 2: Schematic diagram of the cross-linked wormlike loop in the weakly bending regime, which can arise under strong tension or, for large persistence length, under moderately strong tension.
• Figure 3: Schematic diagram of the polymer necklace under strong tension. The two chains are interconnected by multiple reversible cross-links, each of which can be either bound (blue solid line) or unbound (blue dotted line). Cross-linking sites are regularly spaced along the contour at positions $N_k=kN_0$ for the freely jointed necklace (FJN) and $s_k=kL_0$ for the wormlike necklace (WLN), where $N_0$ and $L_0$ are the respective interval parameters for each model.
• Figure 4: Schematic illustration of the Gaussian slinky representation, highlighting the statistical equivalence between cross-linking events in the strongly stretched necklace (left) and loop-formation events in its transverse-plane projection (the Gaussian slinky, right). The matched numbering indices are included in the two panels to emphasize the counting equivalence, and distinct colors are used in the Gaussian slinky to represent different loop sizes for visual distinguishability.
• Figure 5: Graphical solution of the equation $\mathscr{G}_{\rm{B}}(q)=1/\mathscr{G}_{\rm{A}}(q)$ for different values of the statistical weight $w$. Each increasing curve shows $\mathscr{G}_{\rm{B}}(q)$, and each decreasing curve shows $1/\mathscr{G}_{\rm{A}}(q)$. The intersection points (empty diamonds) give the dominant singularity $q^{*}(w)$ of the total generating function $\mathscr{G}(q)$. (a) For $w<1$, $\delta(w)=1-q^{*}(w)$ measures the deviation from the singularity of $\mathscr{G}_{\rm{B}}(q)$ at $q_{\rm{B}}^{*}=1$ (dotted line). As $w$ decreases, $q^{*}(w)\to1$ with $\delta(w)\ll1$. (b) For $w>1$, $\delta(w)=q_{\rm{A}}^{*}-q^{*}(w)$ measures the deviation from the singularity of $\mathscr{G}_{\rm{A}}(q)$ at $q_{\rm{A}}^{*}=1/w$ (dotted lines). As $w$ increases, $q^{*}(w)\to1/w$ with $\delta(w)\ll1$.